Discrete-mode formalism for a field generated by parametric down conversion with an incoherent pump

Abstract

A field generated by parametric down conversion (PDC) with a pump which has a finite coherence time is formalized. This formalism is based on a transformation from continuous mode to discrete mode. The continuous-mode squeezing operator for the PDC output is expressed as a simple product of single-mode and two-mode squeezing operators. It is shown that the derived density operator describing the PDC output does not satisfy a squeezing condition with respect to quadrature amplitudes defined in a standard form. However, it is also shown that the total state composed of the states of the PDC output and a local oscillator (LO) for balanced homodyne detection (BHD) is squeezed with respect to a Hermitian operator, which corresponds to a physical quantity measured by the BHD, including operators of both the PDC output and the LO.